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Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context

Author

Listed:
  • M. D. Ruiz-Medina

    (University of Granada)

  • V. V. Anh

    (Queensland University of Technology)

  • R. M. Espejo

    (University of Granada)

  • J. M. Angulo

    (University of Granada)

  • M. P. Frías

    (University of Jaén)

Abstract

This paper derives conditions under which a stable solution to the least-squares linear estimation problem for multifractional random fields can be obtained. The observation model is defined in terms of a multifractional pseudodifferential equation. The weak-sense and strong-sense formulations of this problem are studied through the theory of fractional Sobolev spaces of variable order, and the spectral theory of multifractional pseudodifferential operators and their parametrix. The Theory of Reproducing Kernel Hilbert Spaces is also applied to define a stable solution to the direct and inverse estimation problems. Numerical projection methods are proposed based on the construction of orthogonal bases of these spaces. Indeed, projection into such bases leads to a regularization, removing the ill-posed nature of the estimation problem. A simulation study is developed to illustrate the estimation results derived. Some open research lines in relation to the extension of the derived results to the multifractal process context are also discussed.

Suggested Citation

  • M. D. Ruiz-Medina & V. V. Anh & R. M. Espejo & J. M. Angulo & M. P. Frías, 2015. "Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 888-911, December.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:3:d:10.1007_s10957-013-0423-4
    DOI: 10.1007/s10957-013-0423-4
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    References listed on IDEAS

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    1. Antoine Ayache & Jacques Vehel, 2000. "The Generalized Multifractional Brownian Motion," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 7-18, January.
    2. Ruiz-Medina, M.D., 2011. "Spatial autoregressive and moving average Hilbertian processes," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 292-305, February.
    3. Ruiz-Medina, M. D. & Angulo, J. M. & Anh, V. V., 2003. "Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 192-216, April.
    4. Fernández-Pascual, Rosaura & Ruiz-Medina, María D. & Angulo, José M., 2006. "Estimation of intrinsic processes affected by additive fractal noise," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1361-1381, July.
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